AP Calculus

Special Focus: The Fundamental
Theorem of Calculus
9 a.m. and time t. Since nobody was in the park when it opened, H(t) therefore gives the
number of people in the park at time t.
Another question similar in spirit to 2000 AB4 and 2002 AB2/BC2 is 2003 (Form B) AB2.
In this question, the student is given the rate at which heating oil is pumped into a tank
and the rate at which heating oil is removed from the tank. Questions are asked about how
many gallons of heating oil are pumped into the tank over a given time interval and how
many gallons are in the tank at a particular time. Both questions can be answered using
the FTC. Also see 2002 (Form B) AB2/BC2, 2004 AB1/BC1, and 2004 (Form B) AB2.
2002 AB3
An object moves along the x-axis with initial position x(0) = 2. The velocity of the object
⎛ π ⎞
at time t ≥ 0 is given by v( t) = sin t
⎝
⎜
⎠
⎟ .
3
(d) What is the position of the object at time t = 4?
Here is a straightforward and common application of the FTC where we are given
an initial position and the velocity function (rate of change of position). Because the
question asks only for the position at a specific time and not the position function, a
general antiderivative is not needed. Rather, the FTC can be used directly to find that
4
4
4 ⎛ π ⎞
x( 4) = x( 0) + ∫ x′ ( t) dt = x( 0) + v( t) dt = 2 + sin
0 ∫
t dt
0 ∫
⎝
⎜ 3 ⎠
⎟ =
0
3. 432,
where the evaluation is done on the calculator. This is a common type of question on the
AP Exam.
2002 AB4/BC4
The graph of the function f shown to the right consists of two
x
line segments. Let g be the function given by g( x) = ∫ f ( t) dt.
0
(a) Find g( −1 ), g ′( −1 ), and g ′′( −1).
(b) For what values of x in the open interval (–2, 2) is g increasing?
Explain your reasoning.
(c) For what values of x in the open interval (–2, 2) is the graph
of g concave down? Explain your reasoning.
AP® Calculus: 2006–2007 Workshop Materials 79